Theorem 0nnq 10926

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5697	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10914	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4037	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3934	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 200	1 ⊢ ¬ ∅ ∈ Q

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146  ∀wral 3081  ∅c0 4286   class class class wbr 5111   × cxp 5661  ‘cfv 6540  2^nd c2nd 7991  Ncnpi 10846   <_N clti 10849   ~_Q ceq 10853  Qcnq 10854

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669  df-nq 10914

This theorem is used by:  adderpq  10958  mulerpq  10959  addassnq  10960  mulassnq  10961  distrnq  10963  recmulnq  10966  recclnq  10968  ltanq  10973  ltmnq  10974  ltexnq  10977  nsmallnq  10979  ltbtwnnq  10980  ltrnq  10981  prlem934  11035  ltaddpr  11036  ltexprlem2  11039  ltexprlem3  11040  ltexprlem4  11041  ltexprlem6  11043  ltexprlem7  11044  prlem936  11049  reclem2pr  11050